A uniform thin rod AB of length L has linear mass density μ(x)=a+bx/L, where x is measured from A. If the CM of the rod lies at a distance of (7/12)L from A, then a and b are related as
a=2b
3a=2b
2a=b
a=b
Solution:
CM of the rod of length l: xcm = ∫0L μ(x)xdx / ∫0L μ(x)dx = ∫0L (a+bx/L)xdx / ∫0L (a+bx/L)dx = aL2/2 + bL2/3 / aL + bL/2 => 7/12L = aL2/2 + bL2/3 / aL + bL/2 => 7/4 = 3a + 2b / 2a + b => 14a + 7b = 12a + 8b => 2a = b